Phase Equilibria in the Grand Canonical Ensemble

A grand-canonical Monte Carlo (GCMC) simulation for a one-component system is performed as follows. The simulation cell has a fixed volume V, and periodic 

Histograms for two runs at different chemical potentials are presented in Fig. 10.2. There is a range of N over which the two runs overlap. In Fig. 10.3 we show the function lnp(iV) — f S/iN for the data of Fig. 10.2. Rearranging (10.15) and taking the logarithm, we see that this function is related to the Helmholtz free energy 

The raw curves for //, i and /.to as well as a composite curve formed by shifting data for the two runs by the amount indicated by the arrows are shown in Fig. 10.3. The combined curve provides information over the combined range of particle numbers, N, covered by the two runs. Note that by keeping one-dimensional histograms for N we are restricted to combining runs of the same temperature, while the more general form (10.14) allows combination of runs at different temperatures. 

As before, our simulation data are subject to statistical (sampling) uncertainties, which are particularly pronounced near the extremes of particle numbers and energies visited during a run. When data from multiple runs are combined, as shown in Fig. 10.3, the question arises of how to determine the optimal amount by which to shift the raw data in order to obtain a global free energy function. As reviewed in 

In both of these equations, we have used a summation rather than an integral over the potential energy for both clarity and the connection with the actual, discretized measurements in a simulation. 

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